Benders Decomposition

Transcription

1 Benders Decomposton John Hooker Carnege Mellon Unversty CP Summer School Cork, Ireland, June 206

2 Outlne Essence of Benders decomposton Smple eample Logc-based Benders Inference dual Classcal LP dual Classcal Benders Eamples 2

3 Outlne Eamples Logc crcut verfcaton Plannng and dsjunctve schedulng Plannng and cumulatve schedulng Mn cost Mn makespan Mn number of late tasks Mn total tardlness Sngle-resource schedulng Home hospce care Branch and check Inference as projecton 3

4 Essence of Benders Decomposton The clever dea behnd classcal Benders works n a much more general settng. For problems that smplfy when certan varables are fed. Use classcal Benders f the resultng subproblem s a lnear programmng (LP) problem.* Same dea can be etended to any subproblem by generalzng LP dualty to nference dualty. * Generalzed Benders allows a nonlnear programmng subproblem 4

5 Essence of Benders Decomposton Master problem Solve for search varables Contans Benders cuts so far generated. F search varables Add Benders cut Subproblem Smplfed problem contans remanng varables y Solve nference dual to obtan Benders cut that ecludes solutons no better than current one.

6 Essence of Benders Decomposton The key to generalzng Benders s generalzng the dual. A soluton of the nference dual s a proof of optmalty (or nfeasblty). It proves a bound on the optmal value Gven the values of search varables as premses. It s an eplanaton of why the soluton s optmal. The same proof may yeld a bound for other values of the search values. Ths s key to obtanng Benders cuts. 6

7 Smple Eample Home $00 $200 Cty 3 Cty 4 Fnd cheapest route to a remote vllage $00 $200 Cty 2 Cty Hgh Pass By ar By bus Vllage 7

8 Smple Eample Let = flght destnaton y = bus route Fnd cheapest route (,y)

9 Let = flght destnaton y = bus route Fnd cheapest route (,y) Master problem Solve for cheapest flght subject to Benders cuts generated so far F flght Add Benders cut Subproblem Fnd cheapest bus route from arport to vllage. Use proof of optmalty to bound cost of other flghts.

10 Let = flght destnaton y = bus route Fnd cheapest route (,y) Begn wth = Cty and pose the subproblem: Fnd the cheapest route gven that = Cty. Optmal cost s $ = $330.

11 The dual problem of fndng the optmal route s to prove optmalty. The proof s that the route from Cty to the vllage must go through Hgh Pass. So cost arfare + bus from cty to Hgh Pass + $50 But ths same argument apples to Cty, 2 or 3. Ths gves us the above Benders cut.

12 Specfcally the Benders cut s cost B ( ) Cty $ $ $00 f f f Cty Cty 2,3 Cty 4 2

13 Now solve the master problem: Pck the cty to mnmze cost subject to cost B ( ) Cty $ $ $00 f f f Cty Cty 2,3 Cty 4 Clearly the soluton s = Cty 4, wth cost $00. 3

14 Now let = Cty 4 and pose the subproblem: Fnd the cheapest route gven that = Cty 4. Optmal cost s $ = $350. $250 4

15 Agan solve the master problem: Pck the cty to mnmze cost subject to cost B ( ) Cty $ $ $00 f f f Cty Cty 2,3 Cty 4 cost B 4( ) Cty $350 $0 f Cty otherwse The soluton s = Cty, wth cost $330. Because ths s equal to the value of a prevous subproblem, we are done. 5

16 Logc-Based Benders Solve problem of the form Iteraton k : Master problem mn f (, y) (, y) S Subproblem mn z z B ( ), k Mnmze cost z subject to Benders cuts Tral value k that solves master Benders cut z B ( ) k k mn f (, y) k (, y) S Solve nference dual to obtan proof of optmalty Use same proof to deduce cost bounds for other assgnments, yeldng 6 Benders cut.

17 Logc-Based Benders In any teraton, master value optmal value smallest subproblem value so far Contnue untl equalty s obtaned. Master problem Subproblem mn z z B ( ), k Mnmze cost z subject to Benders cuts Tral value k that solves master Benders cut z B ( ) k k mn f (, y) k (, y) S Solve nference dual to obtan proof of optmalty Use same proof to deduce cost bounds for other assgnments, yeldng 7 Benders cut.

18 Logc-Based Benders Benders cuts descrbe projecton of feasble set onto f all cuts are generated. Master problem Subproblem mn z z B ( ), k Mnmze cost z subject to Benders cuts Tral value k that solves master Benders cut z B ( ) k k mn f (, y) k (, y) S Solve nference dual to obtan proof of optmalty Use same proof to deduce cost bounds for other assgnments, yeldng 8 Benders cut.

19 Logc-Based Benders Substantal speedup for many applcatons. Several orders of magntude relatve to state of the art. 9

20 Logc-Based Benders Substantal speedup for many applcatons. Several orders of magntude relatve to state of the art. Some applcatons: Crcut verfcaton Chemcal batch processng (BASF, etc.) Steel producton schedulng Auto assembly lne management (Peugeot-Ctroën) Automated guded vehcles n fleble manufacturng Allocaton and schedulng of multcore processors (IBM, Toshba, Sony) Resource locaton-allocaton Stochastc resource locaton and fleet management Capacty and dstance-constraned plant locaton 20

21 Logc-Based Benders Some applcatons Transportaton network desgn Traffc dverson around blocked routes Worker assgnment n a queung envronment Sngle- and multple-machne allocaton and schedulng Permutaton flow shop schedulng wth tme lags Resource-constraned schedulng Wreless local area network desgn Servce restoraton n a network Optmal control of dynamcal systems Sports schedulng 2

22 Inference Dual An optmzaton problem mnmzes an objectve functon subject to constrants. It s solved by searchng over values of the varables. The nference dual fnds the tghtest lower bound on the objectve functon that s mpled by the constrants. It s solved by searchng over proofs.

23 Inference Dual Prmal problem: optmzaton mn f( ) S Fnd best feasble soluton by searchng over values of. ma v P S f ( ) v P Dual problem: Inference P Fnd a proof of optmal value v* by searchng over proofs P. 23

24 Inference Dual Weak dualty always holds: Mn value of prmal problem Ma value of dual problem Dfference = dualty gap 24

25 Inference Dual Strong dualty sometmes holds: Mn value of prmal problem = Ma value of dual problem P s a complete proof famly Strong dualty Complete means that the famly contans a proof for anythng that s mpled by the constrant set. 25

26 Prmal problem mn c A 0 b Classcal LP Dual Inference dual ma v A b P c 0 P P v 26

27 Prmal problem mn c A 0 b Classcal LP Dual Inference dual ma v A b P c 0 P P v Proof famly P : A b P c v 0 when Assumng A b, 0 s feasble. ua ub domnates c for some u 0 v 27

28 Prmal problem mn c A 0 b Classcal LP Dual Inference dual ma v A b P c 0 P P v Proof famly P : A b P c v 0 when Assumng A b, 0 s feasble. ua ub domnates c for some u 0 ua c ub v v 28

29 Prmal problem mn c A 0 b Classcal LP Dual Inference dual ma v A b P c 0 P P v Proof famly P : A b P c v 0 when Assumng A b, 0 s feasble. Ths s a complete nference method (due to Farkas Lemma) ua ub domnates c for some u 0 ua c ub v v 29

30 Classcal LP Dual Prmal problem mn c A 0 b Inference dual ma v A b P c 0 P P v ma v ua c ub v u 0 Proof famly P : A b P c v 0 when Assumng A b, 0 s feasble. Ths s a complete nference method (due to Farkas Lemma) ua ub domnates c for some u 0 ua c ub v v 30

31 Classcal LP Dual Prmal problem mn c A 0 b Inference dual ma ub ua c u 0 ma v ua c ub v u 0 Proof famly P : A b P c v 0 when Assumng A b, 0 s feasble. Ths s a complete nference method (due to Farkas Lemma) ua ub domnates c for some u 0 ua c ub v v 3

32 Classcal LP Dual Prmal problem mn c A 0 b Classcal LP dual ma ua u ub c 0 A strong dual due to Farkas Lemma assumng A b, 0 s feasble Proof famly P : A b P c v 0 when Assumng A b, 0 s feasble. Ths s a complete nference method (due to Farkas Lemma) ua ub domnates c for some u 0 ua c ub v v 32

33 Inference Duals Problem Inference Method Inference dual Lnear programmng Lnear combnaton + domnaton Classcal LP dual (strong) Inequalty constraned optmzaton Lnear combnaton + mplcaton Surrogate dual Inequalty constraned optmzaton Lnear combnaton + domnaton Lagrangean dual Integer programmng Chvátal-Gomory cuts Subaddtve dual (strong) 33

34 Classcal Benders Solve problem of the form Iteraton k : Master problem mn c dy A By b y, 0 Subproblem mn z z B ( ), k Tral value k Benders cut z B ( ) k mn c dy By b A y 0 k k 34

35 Classcal Benders Solve problem of the form Iteraton k : Master problem mn c dy A By b y, 0 Subproblem mn z z B ( ), k Tral value k Benders cut z B ( ) k mn c dy By b A y 0 k k Dual soluton u k proves optmalty: u k By u k ( b A k ) domnates dy v * 35

36 Classcal Benders Solve problem of the form Iteraton k : Master problem mn c dy A By b y, 0 Subproblem mn z z B ( ), k Tral value k Benders cut z B ( ) k mn c dy By b A y 0 k k Dual soluton u k proves optmalty: So k u B d and k k u ( b A ) v * u k By u k ( b A k ) domnates dy v * 36

37 Classcal Benders Solve problem of the form Iteraton k : Master problem mn c dy A By b y, 0 Subproblem mn z z B ( ), k Tral value k Benders cut z B ( ) k mn c dy By b A y 0 k k Dual soluton u k proves optmalty: So k u B d and k k u ( b A ) v * u k By u k ( b A k ) domnates dy v * But u k remans dual feasble for any, so by weak dualty k u ( b A) v 37

38 Classcal Benders Solve problem of the form Iteraton k : Master problem mn c dy A By b y, 0 Subproblem mn z z B ( ), k Tral value k Benders cut z B ( ) k mn c dy By b A y 0 k k Dual soluton u k proves optmalty: So k u B d and k k u ( b A ) v * u k By u k ( b A k ) domnates dy v * But u k remans dual feasble for any, so by weak dualty k Ths mples c u ( b A) c v z k u ( b A) v 38

39 Classcal Benders Solve problem of the form Iteraton k : Master problem mn c dy A By b y, 0 Subproblem mn z k z c u ( b A), k Tral value k Benders cut k z c u ( b A) mn c dy By b A y 0 k k Dual soluton u k proves optmalty: So k u B d and k k u ( b A ) v * u k By u k ( b A k ) domnates dy v * But u k remans dual feasble for any, so by weak dualty k Ths mples c u ( b A) c v z k u ( b A) v 39

40 Classcal Benders Benders s often referred to as row generaton. as opposed to column generaton. Row generaton s much more general. Apples to any optmzaton problem wth constrants = rows Column generaton requres columns. The constrant set must be lnear (A b, etc.)

41 Classcal Benders Benders s often referred to as row generaton. as opposed to column generaton. Row generaton s much more general. Apples to any optmzaton problem wth constrants = rows Column generaton requres columns. The constrant set must be lnear (A b, etc.) Benders s sad to be dual to Dantzg-Wolfe decomposton (a form of column generaton) True for classcal Benders. Not true for logc-based Benders. Logc-based Benders s much more general than D-W or column generaton D-W apples only to lnear programmng.

42 Eample: Logc crcut verfcaton Logc crcuts A and B are equvalent when the followng crcut s a tautology: nputs 2 A and 3 B The crcut s a tautology f the mnmum output over all 0- nputs s. 42

43 For nstance, check whether ths crcut s a tautology: and y not or y 4 nputs 2 not not or y 2 y 3 not or y and 6 y 5 3 and The subproblem s to mnmze the output when the nput s fed to a gven value. Mnmum output s only feasble output, proved by unt propagaton. 43

44 Formally, the problem s mn y6 s.t. y ( y y ) and y not or y y ( y y ) y ( y y ) 4 2 y ( ) y ( ) y ( ) 2 2 not not or y 2 y 3 not or y and 6 y 5 3 and 44

45 mn Master problem z s.t. z B ( ), k Tral nput k Benders cut z B ( ) Only one feasble soluton, trval to compute by unt propagaton k mn Subproblem y 6 s.t. y ( y y ) y ( y y ) y ( y y ) 4 2 k k k 3 y ( ) k k y ( ) 2 3 k k y ( ) 2 45

46 For eample, let the nputs be = (,0,). and 0 y not not or and 0 or y not 4 y 2 not y 3 or y and 6 y 5 To construct a Benders cut, dentfy some nputs that are suffcent to derve an output of by the same unt propagaton. Ths can be done by reasonng backward. 46

47 and 0 y For ths, t suffces that y 4 = and y 5 = not not or and 0 or y not 4 y 2 y 3 not or y and 6 y 5 47

48 For ths, t suffces that y 2 = 0. and 0 y For ths, t suffces that y 4 = and y 5 = not not or and 0 or y not 4 y 2 y 3 not or y and 6 y 5 48

49 For ths, t suffces that y 2 = 0. and 0 y For ths, t suffces that y 4 = and y 5 = not not or and 0 or y not 4 y 2 y 3 not or y and 6 y 5 For ths, t suffces that y 2 = 0. 49

50 For ths, t suffces that 2 = 0 and 3 =. For ths, t suffces that y 2 = 0. and 0 y For ths, t suffces that y 4 = and y 5 = not not or and 0 or y not 4 y 2 y 3 not or y and 6 y 5 For ths, t suffces that y 2 = 0. z So, Benders cut s

51 Now solve the master problem mn s.t. z z 2 3 One soluton s (,, ) (,0,0), z Ths produces output 0 n the net subproblem, at whch pont master and subproblem values converge. Snce mnmum output s 0, crcut s not a tautology. 5

52 Now solve the master problem mn s.t. z z 2 3 One soluton s (,, ) (,0,0), z Ths produces output 0 n the net subproblem, at whch pont master and subproblem values converge. Snce mnmum output s 0, crcut s not a tautology. Note: Ths can also be solved by classcal Benders. The subproblem can be wrtten as an LP (a Horn-SAT problem). 52

53 Eample: Plannng & Schedulng Assgn tasks to resources. Schedule tasks assgn to each resource Subject to tme wndows No overlap (dsjunctve schedulng) Approprate objectve Mn assgnment cost Mn makespan Mn number of late tasks Mn total tardness 53

54 Eample: Plannng & Schedulng Assgn tasks n master, schedule n subproblem. Can combne med nteger programmng and constrant programmng Master problem Subproblem Assgn tasks to resources to mnmze cost. Solve by med nteger programmng. Tral assgnment Benders cut z B ( ) k Schedule tasks on each resource, subject to tme wndows. Advantage: decouples by resource. 54

55 Eample: Plannng & Schedulng Objectve functon Suppose cost s based on task assgnment only. cost c, f task j assgned to resource j j j j So cost appears only n the master problem. Schedulng subproblem s a feasblty problem. 55

56 Eample: Plannng & Schedulng Objectve functon Suppose cost s based on task assgnment only. cost c, f task j assgned to resource j j j j So cost appears only n the master problem. Schedulng subproblem s a feasblty problem. Benders cuts They have the form jj ( ), all j where J s a set of tasks that create nfeasblty when assgned to resource. 56

57 Eample: Plannng & Schedulng Tme wndow relaaton For well-chosen tme ntervals [a,b], jj ( a, b) p b a, all j j p j = processng tme of task j on resource J(a,b) = { tasks wth tme wndows n [a,b] } 57

58 Eample: Plannng & Schedulng Resultng Benders decomposton: Master problem mn z z cj j Benders cuts j Relaaton Tral assgnment Benders cuts jj ( ), j Subproblem Schedule jobs on each resource. For each nfeasble resource, fnd subset J of tasks that create nfeasblty. Termnate when subproblem s feasble. 58

59 Eample: Plannng & Schedulng Problem: We typcally don t have access to nfeasblty proof n subproblem solver. So begn wth smple nogood cut jj ( ), all where J contans all tasks assgned resource. j Then strengthen cut by heurstcally removng tasks from J untl schedule on resource becomes feasble. 59

60 Problem Instances c nstances Hard for LBBD. Some resources much faster than others. Computatonal bottleneck on fastest resource. e nstances Perhaps more realstc. Resources dffer by factor of 2 n processng speed. 60

61 Epermental Desgn Solve wth LBBD Strong Benders cuts only Strengthened nogood cuts. Weak cuts wth subproblem relaaton n master. Smple nogood cuts. Strong cuts wth relaaton. 6

62 Number of nstances solved Performance profle All 80 c nstances Rela + strong cuts Rela + weak cuts Strong cuts only MILP (CPLEX) Computaton tme (sec) 62

63 Number of nstances solved Performance profle 20 c nstances wth 3 or 4 resources Rela + strong cuts Rela + weak cuts Strong cuts only MIP (CPLEX) Computaton tme (sec) 63

64 Number of nstances solved Performance profle 50 e nstances Rela + strong cuts Rela + weak cuts MIP (CPLEX) Computaton tme (sec) 64

65 Severe mbalance of master and subproblem tme, resultng n poorer performance for LBBD. c nstances, 2 resources 65

66 Subproblem blows up when more than 0 tasks per resource on average c nstances, 2 resources 66

67 Subproblem blows up when more than 0 tasks per resource on average c nstances, 3 resources 67

68 Balance between master and subproblem results n superor performance e nstances 68

69 Mld mbalance results n somewhat worse performance e nstances 69

70 Suggested Soluton Strateges Tghter subproblem relaatons Desgn tghter subproblem relaatons for the master usng subproblem varables, whose values are dscarded after master s solved Subproblem decomposton Solve subproblem wth LBBD when t grows too large. More dual nformaton Use subproblem solver that reveals proof of optmalty, perhaps resultng n stronger Benders cuts. 70

71 Cumulatve Schedulng Problems p j = processng tme of task j on resource c j = resource consumpton of task j on resource C = resources avalable on resource Resource Resource 2 C task c task 4 task 5 C 2 c 22 task 3 task 2 p p 22 Total resource consumpton C at all tmes. 7

72 mn Mn Cost Cumulatve Schedulng Master Problem: Assgn tasks to resources Formulate as MILP problem z subject to, all j j jj ( b, a) p c C ( b a), all, varous [ a, b] j j j Benders cuts cost C task task 4 task 5 Relaaton of subproblem: Energy of tasks must be at most energy avalable. a b 72

73 Mn Cost Cumulatve Schedulng Benders cuts same as for dsjunctve schedulng mn z subject to, all j j jj ( b, a) p c C ( b a), all, varous [ a, b] j j j Benders cuts cost C task task 4 task5 Relaaton of subproblem: Energy of tasks must be at most energy avalable. a b 73

74 Mn Makespan Cumulatve Schedulng Master Problem: Assgn tasks to resources Formulate as MILP problem mn subject to M j M C Benders, all j j p cuts j c makespan j j, all C task task 4 task5 Relaaton of subproblem: Energy of tasks provdes lower bound on makespan. 74

75 Mn Makespan Cumulatve Schedulng Benders cuts are based on: Lemma. If we remove tasks, s from a resource, the mnmum makespan on that resource s reduced by at most s j p j ma js d mn d j js j Assumng all deadlnes d are the same, we get the Benders cut * M M ( ) h j J h j p j Mn makespan on resource n last teraton 75

76 Why does ths work? Assume all deadlnes are the same. Add tasks,,s sequentally at end of optmal schedule for other tasks Case I: resultng schedule meets deadlne Optmal makespan for tasks s+,,m Mˆ Optmal makespan for all tasks M * Mˆ s j p j Feasble makespan for all tasks tasks s+,, m task task s M s * Mˆ pj ˆ j M M * s j p j d Deadlne for all tasks 76

77 Case II: resultng schedule eceeds deadlne Optmal makespan for tasks s+,,m Mˆ Optmal makespan for all tasks M * Mˆ s j p j Makespan no longer feasble tasks s+,, m task task s d Deadlne for all tasks M * d s and Mˆ pj d ˆ j M M * s j p j 77

78 Mn Number of Late Tasks Master problem: Assgn tasks to resources mn L subject to, all j j Benders cuts relaaton of subproblem j {0,} = f task j s assgned to resource 78

79 Benders cuts L Lˆ Lˆ Lˆ h h h L L 0, Lˆ * h * h Mn Number of Late Tasks h Lower bound on # late tasks on resource Mn # late tasks on resource (soluton of subproblem) L all * h jj L 0 h ( * h jj h j ( ), j all ), all 79

80 Benders cuts L Lˆ Lˆ Lˆ h h h L L 0, Lˆ * h * h Mn Number of Late Tasks h Lower bound on # late tasks on resource Mn # late tasks on resource (soluton of subproblem) L all * h jj L 0 h ( * h jj h j ( ), j all ), all subset of J h for whch mn # late tasks s stll L h * (found by heurstc that repeatedly solves subproblem on resource ) 80

81 Benders cuts Mn Number of Late Tasks L Lˆ Lˆ Lˆ h h h L L Lˆ * h * h h L * h 0, all jj L 0 h ( * h jj h j ( ), j Mn # late tasks on resource (soluton of subproblem) all ), all To reduce # late tasks, must remove one of the 0 tasks n J h from resource. 8

82 Benders cuts Mn Number of Late Tasks L Lˆ Lˆ Lˆ h h h L L Lˆ * h * h h L * h 0, all jj L 0 h ( * h jj h j ( ), j Mn # late tasks on resource (soluton of subproblem) all ), all subset of J h for whch mn # late tasks s stll L h * (found by heurstc that repeatedly solves subproblem on resource ) Smaller subset of J h for whch mn # late tasks s L h * (found whle runnng same heurstc) 82

83 Benders cuts Mn Number of Late Tasks L Lˆ Lˆ Lˆ h h h L L Lˆ * h * h h L * h 0, all jj L 0 h ( * h jj h j ( ), j Mn # late tasks on resource (soluton of subproblem) all ), all To reduce # late tasks by more than, must remove one of the tasks n J h from resource. 83

84 Benders cuts Mn Number of Late Tasks L Lˆ Lˆ Lˆ h h h L L Lˆ * h * h h L * h 0, all jj L 0 h ( * h jj h j ( ), j all ), all These Benders cuts are added to the master problem n each teraton h. 84

85 Mn Number of Late Tasks Relaaton of subproblem Lower bound on # late tasks on resource L L C L kj ( d c j k ) p ma { p kj ( d j ) k k k } d j, all j 85

86 L L C Mn Number of Late Tasks Relaaton of subproblem L Lower bound on # late tasks on resource kj ( d c j k ) p ma { p kj ( d j ) k k k } d j, Set of tasks assgned to resource wth deadlne at or before d j all C j task task 2 task 3 d j 86

87 L L C Mn Number of Late Tasks Relaaton of subproblem L Lower bound on # late tasks on resource kj ( d c j k ) p ma { p kj ( d j ) k k k } d j, Set of tasks assgned to resource wth deadlne at or before d j all C j task task 2 task 3 Energy = p c d j 87

88 L L C Mn Number of Late Tasks Relaaton of subproblem L Lower bound on # late tasks on resource kj ( d c j k ) p ma { p kj ( d j ) k k k } d Area of tasks assgned to resource wth deadlne at or before d j j, all C j task task 2 task 3 Energy = p c d j 88

89 Mn Number of Late Tasks Relaaton of subproblem L L C L kj ( d c j k ) p ma { p kj ( d j ) k k Lower bound on (makespan latest deadlne) k } d j, all C j task task 2 task 3 d j 89

90 Mn Number of Late Tasks Relaaton of subproblem L L C L kj ( d c j k ) p ma { p kj ( d j ) k k Lower bound on (makespan latest deadlne) k } d j, all C j task task 2 task 3 Ma processng tme d j 90

91 Mn Number of Late Tasks Relaaton of subproblem L L C L kj ( d c j k ) p ma { p kj ( d j ) k k k } d j, all j Mn # of late jobs on resource 9

92 Mn Number of Late Tasks Relaaton of subproblem Ths relaaton s added to the master problem at the outset. L L C L kj ( d c j k ) p ma { p kj ( d j ) k k k } d j, all j Mn # of late jobs on resource 92

93 Mn Tardness Cumulatve Schedulng Master problem: assgn tasks to resources mn L subject to, all j j Benders cuts relaaton I of subproblem relaaton II of subproblem j {0,} = f task j s assgned to resource 93

94 Mn Tardness Cumulatve Schedulng Benders cuts Lower bound on tardness for resource T Tˆ Tˆ Tˆ h h h T T Tˆ * h 0 h h T T * h 0 h 0, all jj jj h h ( \ Z h ( j ), j ), Mn tardness on resource (soluton of subproblem) all all 94

95 Mn Tardness Cumulatve Schedulng Benders cuts Lower bound on tardness for resource T Tˆ Tˆ Tˆ h h h T T Tˆ * h 0 h h T T * h 0 h 0, all jj jj h h ( \ Z h ( j ), j ), Mn tardness on resource (soluton of subproblem) all all To reduce tardness on resource, must remove one of the tasks assgned to t. 95

96 Mn Tardness Cumulatve Schedulng Benders cuts T Tˆ Tˆ Tˆ h h h T T Tˆ * h 0 h h T T * h 0 h 0, all jj jj h h ( \ Z h ( j ), j ), Mn tardness on resource (soluton of subproblem) all all Set of tasks that can be removed, one at a tme from resource wthout reducng mn tardness. 96

97 Mn Tardness Cumulatve Schedulng Benders cuts T Tˆ Tˆ Tˆ h h h T T Tˆ * h 0 h h T T * h 0 h 0, all jj jj h h ( \ Z h ( j ), j ), all all Set of tasks that can be removed, one at a tme from resource wthout reducng mn tardness. Mn tardness on resource when all tasks n Z h are removed smultaneously. 97

98 Mn Tardness Cumulatve Schedulng Benders cuts T Tˆ Tˆ Tˆ h h h T T Tˆ * h 0 h h T T * h 0 h 0, all jj jj h h ( \ Z h ( j ), j ), all all To reduce tardness below on resource, must remove one of the tasks n J h \ Z h Set of tasks that can be removed, one at a tme from resource wthout reducng mn tardness. 0 T h Mn tardness on resource when all tasks n Z h are removed smultaneously. 98

99 Mn Tardness Cumulatve Schedulng These Benders cuts are added to the master problem n each teraton h T Tˆ Tˆ Tˆ h h h T T Tˆ * h 0 h h T T * h 0 h 0, all jj jj h h ( j ), all ( ), all \ Z h j 99

100 Subproblem relaaton I Lower bound on total tardness for resource T T C T jj ( d k c j ) p j j d k, all k 00

101 Subproblem relaaton I T T C T Lower bound on total tardness for resource jj ( d k c j ) p j j d k, Set of tasks assgned to resource wth deadlne at or before d k all k task task 2 task 3 d k 0

102 Subproblem relaaton I T T C T Lower bound on total tardness for resource jj ( d k c j ) p j j d k Area of tasks assgned to resource wth deadlne at or before d k, all k task task 2 task 3 d k 02

103 Subproblem relaaton I Lower bound on total tardness for resource T T C T jj ( d k c j ) p j j Lower bound on total tardness d k, all k task task 2 task 3 d k 03

104 04 Lemma. Consder a mn tardness problem that schedules tasks,, n on resource, where d d n. The mn tardness T* s bounded below by n k T k T where k j k j j k d c p C T ) ( ) ( and s a permutaton of,, n such that ) ( ) ( () ) ( n n c p c p Subproblem relaaton II

105 05 Eample of Lemma j d j p j c j p j c j C task task 2 task 3 d d 2 d 3 Mn tardness = / 5 8) 6 (5 3 ) ( 0 4 6) (5 3 ) ( 0 3 (5) 3 ) ( d c p c p c p C T d c p c p C T d c p C T 2 4/3 bound on tardness Lower 3 2 T T T

106 Idea of proof For a permutaton of,,n let where T T( ) ( ) n k k k ( ) p j c j d ( ) ( ) ( k ) C j T k Let 0 (),, 0 (n) be order of jobs n any optmal soluton, so that t ) t ( ) and mn tardness s T* 0( 0 n Consder bubble sort on 0 (),, 0 (n) to obtan,,n. Let 0,, S be resultng sequence of permutatons, so that s, s+ dffer by a swap and s (j) = j. 06

107 07 Now we have T T T T T T S s s ) ( ) ( ) ( ) ( * 0 snce ) ( 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( * T d c p C d c p C d p t T n j k j j j j n j k j j j j n j j j j n k j s j s k s k k j s j s n k j s j s k s k k j s j s T T T T T T T T T T 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( swap k and k+ So 0 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( B A b a b A A a T T T T T T s k s k s k s k s s snce A a, B b areas def. of

108 08 From the lemma, we can wrte the relaaton n k T T k k where k j k j j j k d c p C T ) ( ) ( ) ( To lnearze ths, we wrte n k T k T and k k k j k j j j k M d c p C T ) ( ) ( ) ( ) ( k j k j j k d c p C M ) ( ) ( where Wrtng relaaton II

109 Computatonal Results Random problems on 2, 3, 4 resources. Facltes run at dfferent speeds. All release tmes = 0. Mn cost and makespan problems: deadlnes same/dfferent. Tardness problems: random due date parameters set so that a few tasks tend to be late. No precedence or other sde constrants. Makes problem harder. Implement wth OPL Studo 09

110 Mn makespan, 2 resources Average of 5 nstances shown Jobs MILP CP Benders At least one problem n the 5 eceeded 7200 sec (2 hours) 0

111 Mn makespan, 3 resources Average of 5 nstances shown Jobs MILP CP Benders At least one problem n the 5 eceeded 7200 sec (2 hours)

112 Mn makespan, 4 resources Average of 5 nstances shown Jobs MILP CP Benders At least one problem n the 5 eceeded 7200 sec (2 hours) 2

113 Mn makespan, 3 resources Dfferent deadlnes Average of 5 nstances shown Jobs MILP CP Benders At least one problem n the 5 eceeded 7200 sec (2 hours) 3

114 Mn # late tasks 3 resources Smaller problems Tasks CP Tme (sec) MILP Benders Mn # late tasks ? > ? > > >

115 Mn # late tasks 3 resources Larger problems Tasks Tme (sec) MILP Benders Best soluton MILP Benders > () >7200 (2) > () > (3) 2 > (5) 2 >7200 >7200 (6) (6) > () 0 > (2) 0 > (3) > (2) ( ) = optmalty not proved 2

116 Effect of subproblem relaaton 3 resources Mn # late tasks Tasks Tme (sec) wth rela wthout rela

117 Mn total tardness 3 resources Smaller problems Tasks Tme (sec) CP MILP Benders Mn tardness > >7200 > > > > > > >

118 Mn total tardness 3 resources Larger problems Tasks Tme (sec) MILP Benders Best soluton MILP Benders >7200 >7200 (75) (37) >7200 >7200 (20) (40) >7200 >7200 (62) (46) >7200 >7200 (375) (4) > (20) 0 > (57) 0 >7200 >7200 (20) (5) >7200 >7200 (25) (7) ( ) = optmalty not proved 5

119 Effect of subproblem relaaton 3 resources Mn total tardness Tasks Tme (sec) wth rela wthout rela >7200 >7200 6

120 Sngle-Resource Schedulng Apply logc-based Benders to sngle-resource schedulng wth long tme horzons and many jobs. Decompose the problem by assgnng jobs to segments of tme horzon. Segmented problem Jobs cannot cross segment boundares (e.g., weekends). Unsegmented problem Jobs can cross segment boundares.

121 Segmented problem Benders approach s very smlar to that for the plannng and schedulng problem. Assgn jobs to tme segments rather than processors. Benders cuts are the same. segment Jobs do not overlap segment boundares

122 Segmented problem Feasblty Wde tme wndows (ndvdual nstances) 22

123 Segmented problem Feasblty Tght tme wndows (ndvdual nstances) 23

124 Segmented problem Mn makespan Wde tme wndows (ndvdual nstances) 24

125 Segmented problem Mn makespan Tght tme wndows (ndvdual nstances) 25

126 Segmented problem Mn tardness Wde tme wndows (ndvdual nstances) 26

127 Segmented problem Mn tardness Tght tme wndows (ndvdual nstances) 27

128 Unsegmented problem Master problem s more complcated. Jobs can overlap two or more segments. Master problem varables must keep track of ths. Benders cuts more sophstcated. segment Jobs can overlap segment boundares

129 Unsegmented problem Master problem: y jk varables keep track of whether job j starts, fnshes, or runs entrely n segment. jk varables keep track of how long a partal job j runs n segment.

130 Unsegmented problem Feasblty -- ndvdual nstances

131 Unsegmented problem Mn makespan ndvdual nstances

132 Sngle-resource schedulng Segmented problems: Benders s much faster for mn cost and mn makespan problems. Benders s somewhat faster for mn tardness problem.

133 Sngle-resource schedulng Segmented problems: Benders s much faster for mn cost and mn makespan problems. Benders s somewhat faster for mn tardness problem. Unsegmented problems: Benders and CP can work together. Let CP run for second. If t fals to solve the problem, t wll probably blow up. Swtch to Benders for reasonably fast soluton.

134 Home Hospce Care Assgn ades to patents. Schedule and route patent vsts for each ade Subject to tme wndows for ades and vsts Subject to ade qualfcaton requrements Weekly schedule Number of vsts per week specfed for each patent Must be same ade and tme for each vst

135 Home Hospce Care Solve wth Benders decomposton. Assgn ades to patents n master problem. Mamze number of patents served by a gven set of ades. Master Problem Solve wth MIP Benders cut Patent, day assgnments Subproblem Solve wth CP

136 Home Hospce Care Solve wth Benders decomposton. Assgn ades to patents n master problem. Mamze number of patents served by a gven set of ades. Schedule home vsts n subproblem. Cyclc weekly schedule. No vsts on weekends. Master Problem Solve wth MIP Benders cut Subproblem Solve wth CP Patent, day assgnments

137 Home Hospce Care Solve wth Benders decomposton. Assgn ades to patents n master problem. Mamze number of patents served by a gven set of ades. Schedule home vsts n subproblem. Cyclc weekly schedule. No vsts on weekends. Subproblem decouples nto a schedulng problem for each ade and each day of the week. Master Problem Solve wth MIP Benders cut Subproblem Solve wth CP Patent, day assgnments

138 Master problem = f patent j assgned to ade Home Hospce Care = f patent j scheduled = f patent j assgned to ade on day k Requred number of vsts per week

139 For a rollng schedule: Home Hospce Care Schedule new patents, drop departng patents from schedule. Provde contnuty for remanng patents as follows: Old patents served by same ade on same days. F y jk = for the relevant ades, patents, and days.

140 For a rollng schedule: Home Hospce Care Schedule new patents, drop departng patents from schedule. Provde contnuty for remanng patents as follows: Old patents served by same ade on same days. F y jk = for the relevant ades, patents, and days. Alternatve: Also served at same tme. F tme wndows to enforce ther current schedule. Alternatve: served only by same ade. F j = for the relevant ades, patents.

141 Home Hospce Care Benders cuts Use strengthened nogood cuts Fnd a smaller set of patents that create nfeasblty by re-solvng the each nfeasble schedulng problem repeatedly. Reduced set of patents whose assgnment to ade on day k creates nfeasblty

142 Home Hospce Care Aulary cuts based on symmetres. A cut for vald for ade, day k s also vald for ade on other days. Ths gves rse to a large number of cuts. The aulary cuts can be summed wthout sacrfcng optmalty. Orgnal cut ensures convergence to optmum. Ths yelds 2 cuts per ade:

143 Home Hospce Care Subproblem relaaton Include relaaton of subproblem n the master problem. Necessary for good performance. Use tme wndow relaaton for each schedulng problem. Smplest relaaton for ade and day k: Set of patents whose tme wndow fts n nterval [a, b]. Can use several ntervals.

144 Home Hospce Care Ths relaaton s very weak. Doesn t take nto account travel tmes. Improved relaaton. Basc dea: Augment vst duraton p j wth travel tme to (or from) locaton j from closest patent or ade home base. Ths s weak unless most assgnments are fed. As n rollng schedule. We partton day nto 2 ntervals. Mornng and afternoon. Smplfes handlng of ade tme wndows and home bases. All patent tme wndows are n mornng or afternoon.

145 Home Hospce Care Tme wndow relaaton for ade, day k usng ntervals [a,b], [b,c] and where Q k = {patents unassgned or assgned to ade, day k}

146 Home Hospce Care Instance generaton Start wth (suboptmal) soluton for the 60 patents F ths schedule for frst n patents. Schedule remanng 60 n patents Use 8 of the 8 ades to cover new patents As well as the old patents they already cover. Ths puts us near the phase transton.

147 Home Hospce Care

148 Home Hospce Care

149 Home Hospce Care

150 Branch and check Generate Benders cuts at certan nodes of a branchng tree Varables fed so far are search varables. Unfed varables go nto subproblem. Not the same as branch and cut. In branch and cut, the cuts contan unfed varables. In branch and check, the cuts contan fed varables. When to use? When master problem s the bottleneck. Master s solved only once, wth growng constrant set. 50

151 Inference as Projecton Project onto propostonal varables of nterest Suppose we wsh to nfer from these clauses everythng we can about propostons, 2, 3 5

152 Inference as Projecton Project onto propostonal varables of nterest Suppose we wsh to nfer from these clauses everythng we can about propostons, 2, 3 We can deduce 2 3 Ths s a projecton onto, 2, 3 52

153 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem 2 Benders cut from prevous teraton 53

154 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master 2 (, 2, 3 ) = (0,,0) Resultng subproblem 54

155 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master 2 (, 2, 3 ) = (0,,0) Resultng subproblem Subproblem s nfeasble. (, 3 )=(0,0) creates nfeasblty 55

156 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master 2 3 (,2,3) = (0,,0) Benders cut (nogood) Resultng subproblem Subproblem s nfeasble. (, 3 )=(0,0) creates nfeasblty 56

157 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master 2 (, 2, 3 ) = (0,,) 3 Resultng subproblem 57

158 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master 2 (, 2, 3 ) = (0,,) 3 Resultng subproblem Subproblem s feasble 58

159 Inference as Projecton Benders decomposton computes a projecton Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master (, 2, 3 ) = (0,,) Enumeratve Benders cut Resultng subproblem Subproblem s feasble 59

160 Inference as Projecton Benders decomposton computes a projecton Logc-based Benders cuts descrbe projecton onto master problem varables. Current Master problem soluton of master JH and Yan (995) JH (202) (, 2, 3 ) = (0,,) Enumeratve Benders cut Contnue untl master s nfeasble. Black Benders cuts descrbe projecton. Resultng subproblem 60

161 Inference as Projecton Benders cuts = conflct clauses n a SAT algorthm Branch on, 2, 3 frst. 6

162 Inference as Projecton Benders cuts = conflct clauses n a SAT algorthm Branch on, 2, 3 frst. Conflct clauses 62

163 Inference as Projecton Benders cuts = conflct clauses n a SAT algorthm Branch on, 2, 3 frst. Conflct clauses Backtrack to 3 at feasble leaf nodes 63

164 Inference as Projecton Benders cuts = conflct clauses n a SAT algorthm Branch on, 2, 3 frst. Conflct clauses contanng, 2, 3 descrbe projecton 64

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